Method of multi-transmitter and multi-path aoa-tdoa location comprising a sub-method for synchronizing and equalizing the receiving stations

ABSTRACT

Method and system for locating one or more transmitters in the potential presence of obstacles in a network comprising a first receiving station A and a second receiving station B that is asynchronous with A. The method includes the identification of a reference transmitter through an estimation of its direction of arrival AOA-TDOA pair (θ ref ,Δτ ref ) on the basis of the knowledge of the positions of the reference transmitter and of stations A and B, an estimation of the direction of arrival of the transmitter or transmitters and of the reflectors (or estimation of the AOA) on station A, and the correction of the errors of asynchronism between station A and station B by using the reference transmitter and the location of the various transmitters on the basis of each pair (AOAi, TDOAi).

CROSS REFERENCE TO RELATED APPLICATIONS

This application is the U.S. National Phase application under 35 U.S.C.§371 of International Application No. PCT/EP2008/066027, filed Nov. 21,2008, and claims the benefit of French Patent Application No. 0708219,filed Nov. 23, 2007, all of which are incorporated by reference herein.The International Application was published on May 28, 2009 as WO2009/065958.

FIELD OF THE INVENTION

The invention relates to a method and a system making it possible tolocate several transmitters in the presence of reflectors on the basisof several receiving stations with synchronization of the receivingstations.

The invention relates to the location of several transmitters in thepresence of reflectors on the basis of several stations. FIG. 1 gives anexemplary location system with 2 receiving stations with position A₁ andA₂ in the presence of two transmitters with position E₁ and E₂ and areflector at R₁. According to FIG. 1, the station at A_(i) receives thedirect path of the transmitter E_(m) at the incidence θ_(mi0) and thereflected path associated with the reflector R_(j) at the incidenceθ_(mij). Location of the transmitters requires not only the estimationof the incidence angles θ_(mij) (AOA abbreviation of “Angle of Arrival”)but also the estimation of the associated TDOAs or time differences ofarrival τ_(mi) ₁ _(j)−τ_(mi) ₂ _(j) between the stations A_(i1) andA_(i2). FIG. 2 shows that the AOA/TDOA location of a transmitter at E₁with the stations with position A₁ and A₂, consists in firstlyestimating its direction θ so as to form a straight line and then inestimating the time difference of arrival Δτ₁₂ of the signal transmittedbetween the two stations so as to form a hyperbola H. The transmitter isthen situated at the intersection of the straight line D of direction θand of the hyperbola H.

Knowing that a receiving station is composed of one or more receivers,the invention also relates to the processing of antennas which processesthe signals of several transmitting sources on the basis of multi-sensorreception systems. In an electromagnetic context the sensors Ci areantennas and the radio-electric sources propagate in accordance with agiven polarization. In an acoustic context the sensors Ci aremicrophones and the sources are sound sources. FIG. 3 shows that anantenna processing system is composed of a network of sensors receivingsources with different angles of arrival θ_(mp). The elementary sensorsof the network receive the signals from the sources possibly beingeither the direct path transmitted by a transmitter or its reflectedpath with a phase and an amplitude depending in particular on theirangles of incidence and the position of the reception sensors. In FIG. 5is represented a particular network of sensors where the coordinates ofeach sensor are (x_(n),y_(n)). The angles of incidence are parametrizedin 1D by the azimuth θ_(m) and in 2D by the azimuth θ_(m) and theelevation Δ_(m). According to FIG. 4, 1D goniometry is defined bytechniques which estimate solely the azimuth by assuming that the wavesfrom the sources propagate in the plane of the sensor network. When thegoniometry technique jointly estimates the azimuth and the elevation ofa source, it corresponds to 2D goniometry.

BACKGROUND OF THE INVENTION

The main objective of antenna processing techniques is to utilize thespatial diversity which consists in using the position of the antennasof the network to better utilize the differences in incidence and indistance of the sources.

One of the technical problems to be solved in this field is that of thelocation of transmitters consisting in determining their coordinates,which will be envisaged in 2 dimensions or 2D, in the plane and/or in 3dimensions or 3D, in space, on the basis of measurements of AOA and/orTDOA type. Multi-transmitter location requires a transmitter-basedassociation of the parameters of AOA/TDOA type, hence joint estimationof the AOA/TDOA parameters.

The field of AOA estimation in the presence of multi-transmitters andmulti-paths on the basis of a multi-channel receiving station is veryvast. That of TDOA estimation is just as wide as that of AOA with inparticular the techniques according the prior art. However, most of thetime the measurement is performed on the basis of two signals arisingfrom two single-channel stations. These techniques are then not veryrobust in multi-transmitter or multi-path situations. This is why theprior art proposes TDOA techniques making it possible to separate thesources on the basis of a priori knowledge about their cycliccharacteristics.

AOA/TDOA joint estimation has generated a large number of referencessuch as described in the prior art. These works are much more recentthan the previous ones on TDOA and are due essentially to the advent ofcellular radio-communications systems as indicated explicitly indocuments of the prior art. Unlike the previous references for TDOA, theprocesses are performed with multi-channel receiving stations. However,the objective is to carry out the parametric analysis of a multi-pathchannel from a single transmitter E₁ to a multi-channel receivingstation at A₁. The jointly estimated parameters are then the angles ofarrival θ_(11j) and the time deviations τ_(11j)−τ_(11j′) between thepaths of this same transmitter due to reflectors at R_(j) and R_(j′).This kind of system does not make it possible to carry out the locationof the transmitter at E₁ such as is envisaged in FIG. 1, unless thepositions of the reflectors at R_(j) and R_(j′) are known. The jointestimation of the parameters (θ_(11j),τ_(11j)−τ_(11j′)) is very oftenenvisaged on the basis of the knowledge of a pilot signal such as theTSC sequence codes (Training Sequence Code) for GSM (Global SystemMobile) or the spreading codes for signals of CDMA (Code DivisionMultiple Access) type.

TOA (Time Of Arrival) estimation techniques have been envisaged forlocating mobiles in cellular radio-communications systems and forlocating radio-navigation receivers of GPS/GALILEO type for the GlobalPositioning System. These estimation techniques are performed on thebasis of the knowledge of a pilot signal and can be carried out withmulti-channel receiving stations. Location often requires thedemodulation of transmitted signals which returns, for example, theposition of the satellites in GPS/GALILEO and allows location of thereceiver on the basis of the knowledge of the position of the satellitesas well as the estimation of the TOA on each of the satellites. The TOAestimation and location techniques then require an accurate knowledge ofthe operation and characteristics of the radio-navigation orradio-communications system but they do not make it possible to carryout location in the general case without a priori knowledge of systemtype or of signal type.

The location of a transmitter on the basis of the AOA/TDOA parametershas spawned a significant bibliography. These data processing techniquesare generally suited to mono-transmitter situations and sometimesenvisage problems with tracking when the transmitter is in motion orelse one of the receiving stations is intentionally in motion. In thisfield numerous references use Kalman filtering. However, these locationtechniques do not deal with the case of TDOA measurements performed onasynchronous receiving stations. In the prior art the authors propose adirect estimation of the position of the transmitters on the basis ofthe set of signals originating from all the reception channels of allthe stations. In this paper, the authors deal with the problem ofmulti-sources through algorithms known from the prior art. It directlyestimates the positions of the transmitters through an antennaprocessing approach. However, it assumes that all the signals have thesame bandwidth and that the signals originating from the variousstations are synchronous. This approach does not, however, make itpossible to deal with the problem of the multi-paths generated byreflectors and the problem of asynchronism between the various stations.

SUMMARY OF THE INVENTION

In an embodiment, the invention relates to a method for locating one ormore transmitters Ei in the potential presence of obstacles Rp in anetwork comprising at least one first receiving station A and one secondreceiving station B asynchronous with A characterized in that itcomprises at least the following steps:

-   -   The identification of a reference transmitter of known position        E₀ by a calculation of the AOA-TDOA pair (θ_(ref),Δτ_(ref)) on        the basis of the knowledge of the position E0 of the reference        transmitter and of those of the stations at A and B,    -   An estimation of the direction of arrival of the transmitter or        transmitters and of the reflectors (or estimation of the AOA) on        the first station A,    -   The separation of the signals received on the first station A by        spatial filtering in the direction of the source (transmitters        and/obstacles),    -   The separation of the incidences originating from the        transmitters from those originating from the obstacles by        inter-correlating the signals arising from the spatial filtering        at A.    -   The estimation of the time difference of arrival or TDOA of a        source (transmitters and/obstacles) by inter-correlating the        signal of the source (transmitters and/obstacles) received at A        with the signals received on the second receiving station B: for        each transmitter source Ei (or obstacles Rj) a pair (AOA, TDOA)        is then obtained,    -   A synthesis of the measurements of the pairs (AOAi, TDOAi) of        each source (Ei, Rp) so as to enumerate the sources and to        determine the means and standard deviation of their AOA and TDOA        parameters,    -   The determination of the error of synchronism between the        receiving stations A and B by using the reference transmitter        E₀, and then the correction of this error on all the TDOAi of        the pairs (AOAi, TDOAi) arising from the synthesis,    -   The determination of the orientation error of the network at A        by using the reference transmitter E₀, and then the correction        of this error on all the AOAi of the pairs (AOAi, TDOAi) arising        from the synthesis,    -   The location of the various transmitters on the basis of each        pair (AOAi, TDOAi).

BRIEF DESCRIPTION OF DRAWINGS

Other characteristics and advantages of the present invention will bemore apparent on reading the description which follows of an exemplaryembodiment given by way of wholly non-limiting illustration, accompaniedby the figures which represent:

FIG. 1, a location system comprising receiving stations at Ai andtransmitters at E_(m),

FIG. 2, an example of AOA/TDOA location in the presence of atransmitter,

FIG. 3, a diagram of a transmitter propagating toward a network ofsensors,

FIG. 4, the incidences (θ_(m),Δ_(m)) of a source,

FIG. 5, an example of a network of sensors with position (xn, yn),

FIG. 6, a system for location on the basis of the stations A and B inthe presence of several transmitters and paths,

FIG. 7, the distortion of the signal transmitted between the receiversat A and at B,

FIG. 8, the MUSIC criterion in the presence of coherent multi-paths (redcurve) and of non-coherent sources (green curve), for directions ofmulti-paths θ₁₁=100° and θ₁₂=200°, the network of N=5 sensors iscircular with a radius of 0.5λ,

FIG. 9, an exemplary elementary Goniometry method taking situations ofcoherent paths into account,

FIG. 10, an illustration of an elementary AOA-TDOA estimation method,

FIG. 11, a representation of the technique for the AOA-TDOA location ofthe transmitter at the position E, and

FIG. 12, the AOA-TDOA location uncertainty ellipse.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 6 represents a location system according to the inventioncomprising for example the following elements:

-   -   M transmitters E_(m) of unknown positions,    -   P reflectors R_(p) of unknown positions,    -   a multi-channel receiving station at A. The station A comprises        a network of sensors since it affords the goniometry function.        The orientation of the antenna of the goniometer at A is for        example known approximately to within Δθ=15°. This corresponds        to the typical accuracy of a magnetic compass,    -   a single- or multi-channel receiving station at B having at        least one reception sensor,    -   a reference transmitter at E₀ whose position is known. The        signal transmitted by this transmitter possesses a transmit band        of the same order of magnitude as that of the receivers at A and        B.

The various parameters of the location system are given in FIG. 6. Inthis system, one of the objectives of the location is to determine theposition of the M transmitters E_(m) of unknown positions. To summarize,the method according to the invention executes at least the followingsteps:

-   -   A goniometry (or estimation of the AOA) of the transmitters Em        and of the reflectors Rp on the station A,    -   A separation of the signals transmitted by spatial filtering in        the direction of the source (transmitter or reflector),    -   The estimation of the time difference of arrival or TDOA of a        source by inter-correlating the signal of a source at the output        of the spatial filtering at A with the signals received at B:        for each source (transmitter Ei or reflector Rj) an (AOA, TDOA)        pair is obtained. This inter-correlation technique making it        possible to estimate the TDOA, will be performed jointly with        the remote “gauging” of the receivers at B, for example.    -   A synthesis of the measurements of the (AOA, TDOA) pairs of each        source will be performed so as to enumerate the transmitters Em        and the reflectors Rp corresponding to the obstacles and to give        statistics, such as the means and standard deviation associated        with the accuracy of estimation of the AOA and TDOA parameters.    -   The identification of the reference transmitter E0 by an AOA        technique from among the (AOA, TDOA) pairs arising from the        synthesis. Knowing the position of the reference transmitter E₀,        the calculation of the error of synchronism between the stations        A and B and then the correction of this error on all the TDOAs        of the (AOA, TDOA) pairs arising from the synthesis. Knowing the        position of the reference transmitter E₀, the calculation of the        orientation error of station A and then the correction of this        error on all the AOAs of the (AOA, TDOA) pairs arising from the        synthesis.    -   The location of the various transmitters and reflectors on the        basis of each (AOA, TDOA) pair and the establishment of the        uncertainty ellipse on the basis of the measurements of standard        deviation of these parameters for each of the transmitters and        reflectors.    -   The calibration error impacting on the goniometry is known,    -   The number K of time slices of duration T over which a joint        estimation of the AOA-TDOA parameter pairs will be performed is        chosen.

The method implemented by the invention is described in more detailhereinafter.

Modeling of the Signal on Station A

In the presence of M transmitters and P obstacles or reflectors, thesignal received as output from the N sensors at A may be written in thefollowing manner according to FIG. 6.

$\begin{matrix}{{x(t)} = {{\sum\limits_{m = 1}^{M}\; {{a\left( \theta_{md} \right)}{s_{m}\left( {t - \tau_{m}} \right)}}} + {\sum\limits_{p = 1}^{P}\; {{a\left( \theta_{pr} \right)}{b_{p}(t)}}} + {n(t)}}} & (1)\end{matrix}$

where s_(m)(t) is the signal of the m-th transmitter, θ_(md) and θ_(pr)are respectively the directions of arrival of the direct path and of areflected path and τ_(m) is the Time Of Arrival (TOA) of the m-thtransmitter such that

$\begin{matrix}{\tau_{m} = {\frac{{E_{m}A}}{c}.}} & (2)\end{matrix}$

where ∥AB∥ is the distance between the points A and B. The signalb_(p)(t) is associated with the p-th obstacle and satisfies:

$\begin{matrix}{{b_{p}(t)} = {\sum\limits_{m = 1}^{M}{\rho_{mp}{{s_{m}\left( {t - \tau_{mp}} \right)}.}}}} & (3)\end{matrix}$

where τ_(mp) is the TOA of the multi-path of the m-th source such that:

$\begin{matrix}{\tau_{mp} = {\frac{{{E_{m}R_{p}}} + {{R_{p}A}}}{c}.}} & (4)\end{matrix}$

and ρ_(mp) is the attenuation of the multi-path of the m-th sourcecaused by the p-th obstacle. The signal of the direct paths_(m)(t−τ_(m)) is correlated with the signal b_(p)(t) originating fromthe obstacle in the following manner

$\begin{matrix}{r_{mp} = {\frac{{E\left\lbrack {{s_{m}\left( {t - \tau_{m}} \right)}{b_{p}(t)}^{*}} \right\rbrack}}{\sqrt{{E\left\lbrack {{s_{m}\left( {t - \tau_{m}} \right)}}^{2} \right\rbrack}{E\left\lbrack {{b_{p}(t)}}^{2} \right\rbrack}}} = {\frac{\rho_{mp}{r_{s_{m}}\left( {\tau_{mp} - \tau_{m}} \right)}}{\sqrt{{r_{s_{m}}(0)}\left( {\sum\limits_{i = 1}^{M}{{\rho_{ip}}^{2}{r_{s_{i}}(0)}}} \right)}}.}}} & (5)\end{matrix}$

where r_(s) _(m) (τ)=E[s_(m)(t)s_(m)(t−τ)*] is the auto-correlationfunction of the signal s_(m)(t) and r_(mp) is a normalized coefficientbetween 0 and 1 giving the degree of correlation between s_(m)(t−τ_(m))and b_(p)(t). When the passband of the m-th transmitter equals B_(m),the function r_(s) _(m) (τ) can be written

$\begin{matrix}{{r_{s_{m}}(\tau)} = {{\gamma_{m}\frac{\sin \left( {\pi \; B_{m}\tau} \right)}{\pi \; B_{m}\tau}\mspace{14mu} {with}\mspace{14mu} \gamma_{m}} = {{E\left\lbrack {{s_{m}(t)}}^{2} \right\rbrack}.}}} & (6)\end{matrix}$

when M=P=1, the expression for r_(s) ₁ _(b) ₁ may be written in thefollowing manner according to (2)(4)(5)(6)

$\begin{matrix}{{r_{11} = {\frac{{r_{s_{1}}\left( {\tau_{11} - \tau_{1}} \right)}}{r_{s_{1}}(0)} = {\sin \; {c\left( {\pi \frac{B_{1}D_{11}}{c}} \right)}\mspace{14mu} {with}}}}{D_{mp} = {{{E_{m}R_{p}}} + {{R_{p}A}} - {{{E_{m}A}}.}}}} & (7)\end{matrix}$

where sin c(x)=sin(x)/x. When x is small the latter function becomes sinc(x)≈1−x²/6. Under these conditions and according to (5)(7), thecorrelation level r_(mp) depends on the distanceD_(mp):r_(mp)≈1−(πB_(m)D_(mp)/c)²/6 in the following manner:

Inversely D_(mp)=c/(πB_(m))√{square root over (6(1−r_(mp)))}. By usingthe above expressions, the multi-paths can be classed into the followingthree categories:

Decorrelated cases: r_(mp)≈0 such that D_(mp)>c/B_(m)

Correlated cases: 0<r_(mp)<r_(max) such thatc/B_(m)<D_(mp)<c/(πB_(m))√{square root over (6(1−r_(max)))}

Coherent case: r_(mp)>r_(max) such that D_(mp)>c/(πB_(m))√{square rootover (6(1−r_(max)))}

In practice r_(max)=0.9 is a typical correlation value for separatingthe cases of coherent multi-paths from the cases of correlatedmulti-paths. The following chart then gives the inter-path distancelimits for obtaining coherent paths.

CHART 1 Distance limit for obtaining coherent paths B_(m) (MHz) 300 kHz1 MHz 10 MHz Limit distance for <246 m <74 m <7 m obtaining coherentpaths

Modeling of the Signal on Station B

The expression for the signal received on the sensors or receivers ofstation B is similar to that of equation (1). However:

-   -   The angles of incidence of the transmitters Em and of the        obstacles or reflector Rp are different: θ_(md)′ and θ_(pr)′        instead of θ_(md) and θ_(pr)    -   The instants of arrival (TOA) of the transmitters Em and of the        obstacles Rp are different: τ_(m)′ and τ_(mp)′ instead of τ_(m)        and τ_(mp) where

$\begin{matrix}{\tau_{m}^{\prime} = {\frac{{E_{m}B}}{c}\mspace{14mu} {and}\mspace{14mu} \tau_{mp}^{\prime}{\frac{{{E_{m}R_{p}}} + {{R_{p}B}}}{c}.}}} & (8)\end{matrix}$

Moreover, the signal of the direct path at the output of the receivers Bmay be written s_(m)′(t−τ_(m)′). Noting that this signal may be writtens_(m)(t−τ_(m)) at the output of the receivers of station A, thedifference between the signals s_(m)(t) and s_(m)′(t) is due to thedifference in the frequency templates (term known in the art) of thereceivers of the stations A from those of the station B. This distortioncaused by receivers of different nature is illustrated in FIG. 7.

To be more precise, in the presence of M transmitters and P obstacles orreflectors, the signal received as output from the N sensors at B may bewritten in the following manner according to FIG. 6

$\begin{matrix}{{x_{B}(t)} = {{\sum\limits_{m = 1}^{M}{{a\left( \theta_{md}^{\prime} \right)}{s_{m}^{\prime}\left( {t - \tau_{m}^{\prime}} \right)}}} + {\sum\limits_{p = 1}^{P}{{a\left( \theta_{pr}^{\prime} \right)}{b_{p}^{\prime}(t)}}} + {{n_{B}(t)}.}}} & (9)\end{matrix}$

where s_(m)′(t) is the signal of the m-th transmitter, θ_(md)′ andθ_(pr)′ are respectively the directions of arrival of the direct pathand of the reflected path and τ_(m)′ is the Time Of Arrival (TOA) of them-th transmitter, the expression for which is given by equation (8). Thesignal b_(p)′(t) is associated with the p-th obstacle and satisfies

$\begin{matrix}{{b_{p}^{\prime}(t)} = {\sum\limits_{m = 1}^{M}{\rho_{mp}^{\prime}{s_{m}^{\prime}\left( {t - \tau_{mp}^{\prime}} \right)}}}} & (10)\end{matrix}$

where τ_(mp)′ is the TOA of the multi-path of the m-th source ofequation (8) and τ_(mp)′ is the attenuation of the multi-path of them-th source caused by the p-th obstacle (or reflector).

Elementary Modules for AOA and TDOA Estimation

Goniometry or Estimation of the Angle of Arrival AOA

The goniometry or AOA estimation algorithms must process the case ofmulti-transmission. With the objective of additionally taking intoaccount the problem of multi-paths, the method can implement twodifferent algorithms:

-   -   The MUSIC scheme in the absence of coherent multi-paths    -   The coherent MUSIC scheme (More generally, this involves the        auto-calibration algorithm known in the prior art) in the        presence of coherent paths, applied when the MUSIC scheme does        not lead to satisfactory results.

It will be considered that it is necessary to apply the Coherent MUSICscheme when for example the estimated correlation {circumflex over(r)}_(mp) between the paths is larger than r_(max) possibly typicallybeing fixed at 0.9. It will also be decided to apply coherent MUSIC whenthe MUSIC scheme has failed.

On output from the goniometry the sources (transmitters and obstacles)are identified either as direct path or as secondary path by a temporalcriterion: The path in the lead over the others is the direct path.

The MUSIC and Coherent MUSIC algorithms are based on the properties ofthe covariance matrix R_(x)=E[x(t)x(t)^(H)] of the observation vectorx(t) of equation (1) where E[.] is the mathematical expectation and ^(H)the conjugation and transposition operator. According to (1), thecovariance matrix may be written

R _(x) =AR _(s) A ^(H)+σ² I _(N) where R _(s) =E[s(t)s(t)^(H)] andE[n(t)n(t)^(H)]=σ² I _(N)   (11)

And where

$\begin{matrix}{{{s(t)} = {\begin{bmatrix}{s_{1}\left( {t - \tau_{1}} \right)} \\\vdots \\{s_{M}\left( {t - \tau_{M}} \right)} \\{b_{1}(t)} \\\vdots \\{b_{P}(t)}\end{bmatrix}\mspace{14mu} {and}}}\text{}{A = {\begin{bmatrix}{a\left( \theta_{1\; d} \right)} & \ldots & {a\left( \theta_{Md} \right)} & {a\left( \theta_{1\; r} \right)} & \ldots & {a\left( \theta_{\Pr} \right)}\end{bmatrix}.}}} & (12)\end{matrix}$

The two schemes are based on the decomposition into eigenelements ofR_(x) where the vectors e_(k) are the eigenvectors associated with theeigenvalue λ_(k) where (λ₁≧λ₂≧ . . . ≧λ_(N)). K is defined as being therank of the matrix R_(x) such that λ₁≧ . . . ≧λ_(K)≧σ²=λ_(K+1) . . .=λ_(N). The two schemes will be differentiated by:

-   -   The structure of the eigenvectors e_(k) of the signal space        (1≦k≦K)    -   The value K of the rank of the matrix R_(x).

The two schemes have in common that they utilize the orthogonalitybetween the eigenvectors of the signal space (1≦k≦K) and theeigenvectors of the noise space (K+1≦k≦N). The criteria associated withthe two schemes will require the calculation of the noise projectorwhere

$\begin{matrix}{\Pi_{b} = {\sum\limits_{i = {K + 1}}^{N}{e_{i}{e_{i}^{H}.}}}} & (13)\end{matrix}$

In practice the estimate {circumflex over (Π)}_(b) of the noiseprojector is deduced from the following estimate {circumflex over(R)}_(x)(T₀) of the covariance matrix R_(x)

$\begin{matrix}{{{\hat{R}}_{x}\left( T_{0} \right)} = {\frac{1}{T_{0}}{\sum\limits_{t = 1}^{T_{0}}{{x(t)}{{x(t)}^{H}.}}}}} & (14)\end{matrix}$

Case of Non-Coherent Path and Application of MUSIC

In this case the rank of the matrix R_(x) equals K=M+P since thecovariance matrix of the sources R_(s) is of full rank. Under theseconditions, the K eigenvectors of the signal space may be written:

$\begin{matrix}{e_{k} = {\sum\limits_{k = 1}^{K}\; {\alpha_{k}{a\left( \theta_{k} \right)}\mspace{14mu} {for}\mspace{14mu} {\left( {1 \leq k \leq K} \right).}}}} & (15)\end{matrix}$

In this particular case the matrix A is of dimension N×K since A=[a(θ₁). . . a(θ_(K))]. Knowing that the decomposition into eigenelements ofR_(x) induces the orthogonality between the e_(k) of the signal space ofequation (15) and the e_(i) of the noise space of equation (13), thevectors a(θ_(k)) are orthogonal to the columns of the noise projectorΠ_(b). Under these conditions, the incidences θ_(k) of the K sources arethe K minima which cause the following MUSIC criterion to vanish:

$\begin{matrix}{{J_{MUSIC}(\theta)} = {\frac{{a(\theta)}^{H}\Pi_{b}{a(\theta)}}{{a(\theta)}^{H}{a(\theta)}}.}} & (16)\end{matrix}$

The MUSIC criterion J_(MUSIC)(θ) is calculated for θ ranging from 0 to360° and is normalized between 0 and 1 since for all θ it satisfies:0≦J_(MUSIC)(θ)≦1.

In the method two techniques can be implemented for detecting thepresence of coherent sources:

-   -   A detection of coherent sources by thresholding the criterion        J_(MUSIC)(θ).    -   A detection of coherent sources after estimating the level of        correlation between the sources.

Detection of Coherent Sources by Thresholding J_(MUSIC)(θ)

In order to better understand the choice of a threshold the behavior ofthe MUSIC criterion J_(MUSIC)(θ) is simulated in the presence of twopaths with incidences θ₁₁=100° and θ₁₂=200° in the coherent and then thenon-coherent cases. The result of the simulation is given in FIG. 8. Thenetwork of N=5 sensors is circular with a radius of 0.5λ. In thiscontext, good operation of MUSIC is characterized by the fact thatJ_(MUSIC)(θ₁₁=100°) and J_(MUSIC)(θ₁₂=200°) are zero. According to FIG.8 this good property is satisfied when the two multi-paths arenon-coherent. In the coherent case, J_(MUSIC)(θ₁₁=100°) andJ_(MUSIC)(θ₁₂=200°) are on the one hand far from being zero and on theother hand the K=2 smallest minima associated with the estimates{circumflex over (θ)}₁₁ and {circumflex over (θ)}₁₂ of J_(MUSIC)(θ) aremuch further from θ₁₁=100° and θ₁₂=200° than in the coherent case.Moreover in the coherent case the dynamic swing between the K=2 smallestminima of the criterion and the following minimina is very low, andhence there is a significant risk of estimating ambiguous directions ofarrival.

The example of FIG. 8 shows that it is easy to eliminate the poorgoniometries {circumflex over (θ)}₁₁ and {circumflex over (θ)}₁₂ relatedto the presence of coherent sources by a threshold of good goniometry“threshold_(—)1st”. When a minimum J_(MUSIC)({circumflex over (θ)}_(m))satisfies J_(MUSIC)({circumflex over (θ)}_(m))<threshold_(—)1st, theazimuth {circumflex over (θ)}_(m) is the direction of a non-coherentpath and when J_(MUSIC)({circumflex over (θ)}_(m))>threshold_(—)1st, theazimuth {circumflex over (θ)}_(m) is not associated with a direction ofarrival. The presence of coherent paths is then detected when the number{circumflex over (K)} of minima satisfying J_(MUSIC)({circumflex over(θ)}_(k))<threshold_(—)1st is less than the rank K of the covariancematrix R_(x).

Detection of Coherent Sources by Estimating the Inter-Correlation Level

In this case where K=M+P, the K estimated incidences satisfyJ_(MUSIC)({circumflex over (θ)}_(k))<threshold_(—)1st. However, it isknown moreover that the more significant the level of correlationbetween the sources, the larger the variance of the estimates{circumflex over (θ)}_(m). The objective is then to estimate thecovariance matrix of the sources R_(s) of equation (11) on the basis ofthe estimates of the matrix A as well as the noise level σ². On thebasis of the covariance matrix {circumflex over (R)}_(x)(T₀) and of theestimates {circumflex over (θ)}₁ . . . {circumflex over (θ)}_(K), thesteps of the method are as follows:

Step A.1: On the basis of the result of EVD of {circumflex over(R)}_(x)(T₀)=Σ_(i=1) ^(N)λ_(i)e_(i)e_(i) ^(H), which is used toconstruct Π_(b), calculation of an estimate of the noise level

{circumflex over (σ)}²=(Σ_(i=K+1) ^(N)λ₁)/(N−K).   (17)

Step A.2: Calculation of an estimate of the denoised covariance matrixR_(y)=A R_(s) A^(H) by performing

{circumflex over (R)} _(y)=Σ_(i=1) ^(K)(λ_(i)−{circumflex over (σ)}²)e_(i) e _(i) ^(H).   (18)

Step A.3: On the basis of the estimate Â=[a({circumflex over (θ)}₁) . .. a({circumflex over (θ)}_(K))] of the matrix of direction vectors,deduction of an estimate of the covariance matrix of the sources

{circumflex over (R)} _(s) =A ^(#) {circumflex over (R)} _(y)(A^(#))^(H) where A ^(#)=(A ^(H) A)⁻¹ A ^(H).   (19)

Step A.4: Calculation of the maximum correlation {circumflex over(r)}_(max) between the paths i.e.

$\begin{matrix}{{\hat{r}}_{\max} = {\max\limits_{i,j}{\left( \frac{{{\hat{R}}_{s}\left( {i,j} \right)}}{\sqrt{{{\hat{R}}_{s}\left( {i,i} \right)}{{\hat{R}}_{s}\left( {j,j} \right)}}} \right).}}} & (20)\end{matrix}$

The estimated correlation {circumflex over (r)}_(mp) between the pathsis larger than r_(max) possibly typically being fixed at 0.9.

The coherent MUSIC technique will be used when {circumflex over(r)}_(max)>r_(max). A typical value of r_(max) is 0.9.

Case of Coherent Path and Application of Coherent MUSIC

In this case the rank of the matrix R_(x) satisfies K<M+P since thecovariance matrix of the sources R_(s) is no longer of full rank. Underthese conditions, the K eigenvectors of the signal space may be written:

$\begin{matrix}{e_{k} = {\sum\limits_{k = 1}^{K}{\alpha_{k}{b\left( {\theta_{k},\rho_{k},I_{k}} \right)}\mspace{14mu} {for}\mspace{14mu} {\left( {1 \leq k \leq K < {M + P}} \right).}}}} & (21)\end{matrix}$

Where

$\begin{matrix}{{b\left( {\theta_{k},\rho_{k},I_{k}} \right)} = {{\sum\limits_{p = 1}^{I_{k}}\; {\rho_{p}{a\left( \theta_{kp} \right)}}} = {{U_{I_{k}}\left( \theta_{k} \right)}{\rho_{k}.}}}} & (22)\end{matrix}$

with U_(I) _(k) (θ_(k))=[a(θ_(k1)) . . . a(θ_(kI) _(k) )] andρ_(k)=[ρ_(k1) . . . ρ_(kI) _(k) ]^(T)Where the θ_(kp)(1≦p≦I_(k)) are the incidences of the coherent pathsassociated with the same transmitter, with this model Σ_(k=1)^(K)I_(k)=M+P. Knowing that the decomposition into eigenelements ofR_(x) induces the orthogonality between the e_(k) of the signal space ofequation (13) and the e_(i) of the noise space of equation (15), thevectors b(θ_(k),ρ_(k),I_(k)) are orthogonal to the columns of the noiseprojector Π_(b). According to (21)(22) and the coherent MUSIC algorithmof [4], the incidences θ_(k)=[θ_(k1) . . . θ_(kI)] of the K groups ofcoherent sources are the K minima which cause the following coherentMUSIC criterion to vanish

$\begin{matrix}{{J_{MC}\left( {\theta,I} \right)} = {\frac{\det\left( {{U_{I}\left( {\theta,P} \right)}^{H}\Pi_{b}{U_{I}\left( {\theta,P} \right)}} \right)}{\det\left( {{U_{I}\left( {\theta,P} \right)}^{H}{U_{I}\left( {\theta,P} \right)}} \right)}.}} & (23)\end{matrix}$

The MUSIC criterion J_(MC)(θ,I) is calculated for all the I-tuples θ=[θ₁. . . θ_(I)] satisfying θ₁> . . . >θ_(I) where the θ_(i) vary between 0and 360°. The criterion J_(MC)(θ,I) is moreover normalized between 0 and1 since it satisfies 0≦J_(MC)(θ)≦1. Just as for MUSIC the elimination ofthe poor I-tuples will be done by way of a threshold of good goniometry“threshold_(—)1st”. Consequently, to be valid, the I-tuplesθ_(k)=[θ_(k1) . . . θ_(kI)] must satisfy J_(MC)({circumflex over(θ)}_(k))<threshold_(—)1st. If the number of good I-tuples is less thanK, the coherent MUSIC scheme is repeated for I=I+1.

The steps of coherent MUSIC are then as follows:

Step B.1: Initialization to I=2

Step B.2: Calculation of the criterion of equation (23) for all theθ=[θ₁ . . . θ_(I)] satisfying θ₁> . . . >θ_(I) knowing that the θ_(i)vary between 0 and 360°.

Step B.3: Search for the {circumflex over (K)} I-tuples satisfyingJ_(MC)({circumflex over (θ)}_(k))<threshold_(—)1^(st).

Step B.4: If {circumflex over (K)}<K then return to step B.2 with I=I+1.

Step B.5: Calculation of the set {{circumflex over (θ)}₁ . . .{circumflex over (θ)}_(M+P)} of incidences of the sources by calculating{{circumflex over (θ)}₁ . . . {circumflex over (θ)}_(M+P)}=∩_(i=1)^({circumflex over (K)})θ_(k)

Separation of the Direct Paths from the Reflected Paths

The separation of the paths is performed on the basis of the signal x(t)of equation (1) as well as the estimates of the incidences of thesources {{circumflex over (θ)}₁ . . . {circumflex over (θ)}_(M+P)}.According to equations (1) and (12) the signal received can be written:

x(t)=As(t)+n(t).   (24)

Consequently, the vector s(t) is estimated on the basis of an estimateÂ=[a({circumflex over (θ)}₁) . . . a({circumflex over (θ)}_(M+P))] ofthe matrix A as well as the observation vector x(t) by a spatialfiltering technique. By applying a least squares technique

ŝ(t)=(Â ^(H) Â)⁻¹ Â ^(H) x(t).   (25)

The i-th component ŝ_(i)(t) of ŝ(t) can have the following twoexpressions according to (12)

$\begin{matrix}{{{{\hat{s}}_{i}(t)} = {s_{m}\left( {t - \tau_{m}} \right)}}{{{\hat{s}}_{i}(t)} = {{b_{p}(t)} = {\sum\limits_{m = 1}^{M}\; {\rho_{mp}{{s_{m}\left( {t - \tau_{mp}} \right)}.}}}}}} & (26)\end{matrix}$

By considering the following inter-correlation criterion:

$\begin{matrix}{{r_{ij}(\tau)} = {\frac{{{E\left\lbrack {{{\hat{s}}_{i}(t)}{{\hat{s}}_{j}\left( {t - \tau} \right)}} \right\rbrack}}^{2}}{{E\left\lbrack {{{\hat{s}}_{i}(t)}}^{2} \right\rbrack}{E\left\lbrack {{{\hat{s}}_{j}\left( {t - \tau} \right)}}^{2} \right\rbrack}}.}} & (27)\end{matrix}$

For i<j, the following two situations are encountered, knowing that thesignals of the M transmitters are independent

$\begin{matrix}{{{{Case}\mspace{14mu} {n{^\circ}}\mspace{14mu} 1\text{:}\mspace{14mu} {If}\mspace{14mu} {{\hat{s}}_{i}(t)}} = {{{s_{m}\left( {t - \tau_{m}} \right)}\mspace{14mu} {and}\mspace{14mu} {{\hat{s}}_{j}(t)}} = {s_{m^{\prime}}\left( {t - \tau_{m^{\prime}}} \right)}}}{{{then}\mspace{14mu} {r_{ij}(\tau)}} = 0}{{{Case}\mspace{14mu} {n{^\circ}}\mspace{14mu} 2\text{:}\mspace{14mu} {If}\mspace{14mu} {{\hat{s}}_{i}(t)}} = {s_{m}\left( {t - \tau_{m}} \right)}}{{{and}\mspace{14mu} {{\hat{s}}_{j}(t)}} = {{b_{p}(t)} = {\sum\limits_{m^{\prime} = 1}^{M}{\rho_{m^{\prime}p}{s_{m^{\prime}}\left( {t - \tau_{m^{\prime}p}} \right)}\mspace{14mu} {then}}}}}\begin{matrix}{{\max\limits_{\tau}{r_{ij}(\tau)}} = {r_{ij}\left( {\tau_{mp} - \tau_{m}} \right)}} \\{= \frac{{\rho_{mp}}^{2}{E\left\lbrack {{s_{m}(t)}}^{2} \right\rbrack}}{\sum\limits_{m^{\prime} = 1}^{M}{{\rho_{m^{\prime}p}}^{2}{E\left\lbrack {{s_{m^{\prime}}(t)}}^{2} \right\rbrack}}}}\end{matrix}} & (28) \\{{{{Case}\mspace{14mu} {n{^\circ}}\mspace{14mu} 3\text{:}\mspace{14mu} {If}\mspace{14mu} {{\hat{s}}_{i}(t)}} = {b_{p^{\prime}}(t)}}{{{and}\mspace{14mu} {{\hat{s}}_{j}(t)}} = {{b_{p}(t)} = {\sum\limits_{m^{\prime} = 1}^{M}{\rho_{m^{\prime}p}{s_{m^{\prime}}\left( {t - \tau_{m^{\prime}p}} \right)}\mspace{14mu} {then}}}}}\begin{matrix}{{\max\limits_{\tau}{r_{ij}(\tau)}} = {r_{ij}\left( {\tau_{mp} - \tau_{{mp}^{\prime}}} \right)}} \\{= {\frac{{{\rho_{mp}\rho_{{mp}^{\prime}}}}^{2}{E\left\lbrack {{s_{m}(t)}}^{2} \right\rbrack}^{2}}{\left( {\sum\limits_{m^{\prime} = 1}^{M}{{\rho_{{mp}^{\prime}}}^{2}{E\left\lbrack {{s_{m}(t)}}^{2} \right\rbrack}}} \right)\left( {\sum\limits_{m^{\prime} = 1}^{M}{{\rho_{mp}}^{2}{E\left\lbrack {{s_{m}(t)}}^{2} \right\rbrack}}} \right)}.}}\end{matrix}} & \;\end{matrix}$

Consequently, the filtering outputs ŝ_(i)(t) associated with the signalsb_(p)(t) are correlated with all the other filtering outputs ŝ_(j)(t)for 1≦j≦M+P. The signals ŝ_(i)(t) associated with the reflectors (orobstacles) will be those which satisfy

$\mspace{14mu} {{\max\limits_{\tau}{r_{ij}(\tau)}} \neq 0}$

for i<k and the other outputs will be associated with the direct path.

In practice

$\max\limits_{\tau}{r_{ij}(\tau)}$

is compared with a threshold η to decide a correlation between ŝ_(i)(t)and ŝ_(j)(t) (A typical value of η is 0.1). The method for separatingthe direct and reflected paths consisting in identifying the setsΘ_(d)={θ_(1d) . . . θ_(Md)} and Θ_(r)={θ_(1r) . . . θ_(Pr)} is then asfollows:

Step C.0: Θ_(d)= and Θ_(r)=

Step C.1: Construction of the matrix Â=[a({circumflex over (θ)}₁) . . .a({circumflex over (θ)}_(M+P))] of the sources consisting of thetransmitters and the obstacles on the basis of the set of incidences{{circumflex over (θ)}¹ . . . {circumflex over (θ)}_(M+P)} estimatedeither by MUSIC or by Coherent MUSIC.

Step C.2: Estimation of the signal vector ŝ(t) of dimension (M+P)×1 onthe basis of Â and of the sensor signals x(t) by a spatial filteringtechnique. An exemplary spatial filtering is given by equation (25).

Step C.3: Initialization to i=1.

Step C.4: Calculation of

$r_{ij}^{\max} = {\max\limits_{\tau}{r_{ij}(\tau)}}$

for 1≦j≦M+P.

Step C.5: If for 1≦j≦M+P, r_(ij) ^(max)>η then Θ_(r)={θ_(i)}∪Θ_(r).

Step C.6: If for 1≦j≦M+P, there exists at least one value of j such thatr_(ij) ^(max)≦η then Θ_(d)={θ_(i)}∪Θ_(d).

Step C.7: If i<M+P return to step No. C.4 with i=i+1

Method of Goniometry in the Possible Presence of Coherent Multi-Paths

The method of elementary goniometry in the possible presence ofmulti-paths is represented in the diagram of FIG. 9. More precisely thesteps are as follows:

Step D.1: Acquisition of the signal x(t) and correction of thedistortions of the receivers by a gauging process known to the personskilled in the art.

Step D.2: Calculation of the covariance matrix {circumflex over(R)}_(x)(T₀) of equation (14);

Step D.3: Following an eigenvalue decomposition or “EVD” of {circumflexover (R)}_(x)(T₀), determination of the rank K of this matrix andconstruction of the noise projector {circumflex over (Π)}_(b) accordingto equation (13).

Step D.4: Application of MUSIC: Search as a function of θ for the K′minima θ of the criterion J_(MUSIC)(θ) of equation (16) satisfyingJ_(MUSIC)({circumflex over (θ)}_(k))<threshold_(—)1^(st) for 1≦k≦K′. IfK′<K go to step No. D.6.

Step D.5: Calculation of the maximum degree of correlation {circumflexover (r)}_(max) between the sources in accordance with steps No. A.1 toA.4. If {circumflex over (r)}_(max)<r_(max) go to step No. D.7 andconstruction of the set of incidences {{circumflex over (θ)}₁ . . .{circumflex over (θ)}_(M+P)}.

Step D.6: Application of the coherent-MUSIC scheme according to stepsNo. B.1 to B.5 to obtain the set of incidences {{circumflex over (θ)}₁ .. . {circumflex over (θ)}_(M+P)}

Step D.7: Construction of the sets of incidences Θ_(d)={θ_(1d) . . .θ_(Md)} and Θ_(r)={θ_(1r) . . . θ_(Pr)} associated respectively with thedirect paths and with the multi-paths on the basis of the set ofincidences {{circumflex over (θ)}₁ . . . {circumflex over (θ)}_(M+P)}according to steps C.

TDOA (Time Difference of Arrival) Estimation FIG. 10

The objective of this paragraph is to estimate the TDOA τ_(m)−τ_(m)′ ofeach of the direct paths as well as the TDOA τ_(mp)−τ_(mp)′ of thereflectors which according to (2)(4) satisfy

$\begin{matrix}{{{\tau_{mp} - \tau_{mp}^{\prime}} = {\frac{{{R_{p}A}} - {{R_{p}B}}}{c}\mspace{14mu} {and}}}\mspace{14mu} {{\tau_{m} - \tau_{m}^{\prime}} = {\frac{{{E_{m}A}} - {{E_{m}B}}}{c}.}}} & (29)\end{matrix}$

Steps C.1 and C.2 described previously make it possible on the basis ofthe set of incidences {{circumflex over (θ)}₁ . . . {circumflex over(θ)}_(M+P)} to deduce the signal vector ŝ(t) of equation (12). Knowingon the one hand that the i-th component ŝ_(i)(t) of ŝ(t) is the signalassociated with the source of incidence {circumflex over (θ)}_(i) andthat on the other hand according to steps C previously described it ispossible to identify incidences of the direct paths if {circumflex over(θ)}_(i)∈{θ_(1d) . . . θ_(Md)} or of the reflected paths if {circumflexover (θ)}_(i)∈{θ_(1r) . . . θ_(Pr)}. Thus:

$\begin{matrix}{{{{{If}\mspace{14mu} {\hat{\theta}}_{i}} \in {\left\{ {\theta_{1d}\mspace{14mu} \ldots \mspace{14mu} \theta_{Md}} \right\} \mspace{14mu} {then}\mspace{14mu} {{\hat{s}}_{i}(t)}}} = {s_{m}\left( {t - \tau_{m}} \right)}}{{{If}\mspace{14mu} {\hat{\theta}}_{i}} \in {\left\{ {\theta_{1r}\mspace{14mu} \ldots \mspace{14mu} \theta_{\Pr}} \right\} \mspace{14mu} {then}}}\text{}{{{\hat{s}}_{i}(t)} = {{b_{p}(t)} = {\sum\limits_{m = 1}^{M}{\rho_{mp}{{s_{m}\left( {t - \tau_{mp}} \right)}.}}}}}} & (30)\end{matrix}$

The signals x_(B)(t) received on the reception system at B (see FIG. 6)have the expression of equations (9)(10). In the method, the distortionbetween the signals s_(m)(t) and s_(m)′(t) received respectively at Aand B is modeled by the following FIR filter:

$\begin{matrix}{{s_{m}^{\prime}(t)} = {{\sum\limits_{i = {- L}}^{L}{h_{i}{s_{m}\left( {t - {iT}_{e}} \right)}}} = {\underset{\underset{h^{T}}{}}{\begin{bmatrix}h_{- L} & h_{- L}\end{bmatrix}}{\underset{\underset{s_{m}{(t)}}{}}{\begin{bmatrix}{s_{m}\left( {t + {LT}_{e}} \right)} \\\vdots \\{s_{m}\left( {t - {LT}_{e}} \right)}\end{bmatrix}}.}}}} & (31)\end{matrix}$

Consequently the signal x_(B)(t) becomes

$\begin{matrix}{{x_{B}(t)} = {{\sum\limits_{m = 1}^{M}{{a\left( \theta_{md}^{\prime} \right)}h^{T}{s_{m}\left( {t - \tau_{m}^{\prime}} \right)}}} + {\sum\limits_{p = 1}^{P}{{a\left( \theta_{pr}^{\prime} \right)}h^{T}{b_{p}^{\prime}(t)}}} + {{n_{B}(t)}.}}} & (32)\end{matrix}$

Knowing that b_(p)′(t)=h^(T) b_(p)′(t). According to equations (10)(32)

$\begin{matrix}{{b_{p}^{\prime}(t)} = {\sum\limits_{m = 1}^{M}{\rho_{mp}^{\prime}{{s_{m}\left( {t - \tau_{mp}^{\prime}} \right)}.}}}} & (33)\end{matrix}$

And therefore

$\begin{matrix}{{x_{B}(t)} = {{\sum\limits_{m = 1}^{M}{{a\left( \theta_{md}^{\prime} \right)}h^{T}{s_{m}\left( {t - \tau_{m}^{\prime}} \right)}}} + {\sum\limits_{p = 1}^{P}{\sum\limits_{m = 1}^{M}{\rho_{mp}^{\prime}{a\left( \theta_{pr}^{\prime} \right)}h^{T}{s_{m}\left( {t - \tau_{mp}^{\prime}} \right)}}}} + {{n_{B}(t)}.}}} & (34)\end{matrix}$

-   -   Consequently when ŝ_(i)(t)=s_(m)(t−τ_(m)), the difference in the        arrival time or TDOA τ=τ_(m)−τ_(m)′ will correspond to a maximum        correlation between the signals s_(m)(t−τ_(m)) and x_(B)(t+τ).        The multi-channel correlation criterion constructed is based on        Gardner's theory [26][27]

$\begin{matrix}{{{{\hat{c}}_{xy}(\tau)} = {1 - {\det\left( {I_{N} - {{\hat{R}}_{xx}^{- 1}{{\hat{R}}_{xy}(\tau)}{{\hat{R}}_{yy}(\tau)}^{- 1}{{\hat{R}}_{yx}\left( {\tau,f} \right)}}} \right)}}}{{{\hat{R}}_{xy}(\tau)} = {\frac{1}{K}{\sum\limits_{k = 1}^{K}{{x\left( {kT}_{e} \right)}{y\left( {{kT}_{e} + \tau} \right)}^{H}}}}}{{\hat{R}}_{xx} = {\frac{1}{K}{\sum\limits_{k = 1}^{K}{{x\left( {kT}_{e} \right)}{x\left( {kT}_{e} \right)}^{H}}}}}{{{\hat{R}}_{yy}(\tau)} = {\frac{1}{K}{\sum\limits_{k = 1}^{K}{{y\left( {{kT}_{e} + \tau} \right)}{{y\left( {{kT}_{e} + \tau} \right)}^{H}.}}}}}} & (35)\end{matrix}$

With x(t)=s_(m)(t−τ_(m)) and y(t)=x_(B)(t). The TDOA of the m-th sourceis then

$\begin{matrix}{{\tau_{m} - \tau_{m}^{\prime}} = {\max\limits_{\tau}{{{\hat{c}}_{xy}(\tau)}.}}} & (36)\end{matrix}$

According to (34) and in the presence of P obstacles, the functionĉ_(xy)(τ) contains P other maxima in τ_(m)−τ_(mp)′ for 1≦p≦P. Knowingthat τ_(mp)′>τ_(m)′, the method will retain the TDOA τ_(m)−τ_(m)′knowing that it satisfies τ_(m)−τ_(m)′<τ_(m)−τ_(mp)′.

-   -   When ŝ_(i)(t)=b_(p)(t)=Σ_(m−1) ^(M)ρ_(mp) s_(m)(t−τ_(mp)) the        observation vector is constructed

$\begin{matrix}{{b_{p}(t)} = {\begin{bmatrix}{b_{p}\left( {t + {LT}_{e}} \right)} \\\vdots \\{b_{p}\left( {t - {LT}_{e}} \right)}\end{bmatrix} = {\sum\limits_{m = 1}^{M}\; {\rho_{m\; p}{{s_{m}\left( {t - \tau_{m\; p}} \right)}.}}}}} & (37)\end{matrix}$

The TDOA τ_(mp)−τ_(mp)′ will correspond to a maximum correlation betweenthe signals b_(p)(t) and x_(B)(t+τ). The multi-channel correlationcriterion of equation (35) is constructed with x(t)=b_(p)(t) andy(t)=x_(B)(t). The correlation criterion ĉ_(xy)(τ) also contains P othercorrelation maxima in τ_(mp)−τ_(m)′. Knowing that τ_(mp)′>τ_(m)′, themethod will retain the TDOA of interest τ_(mp)−τ_(mp)′ knowing that itsatisfies τ_(mp)−τ_(mp)′<τ_(mp)−τ_(m)′.

According to the above description the method for associating the anglesof arrival and time differences of arrival TDOA is as follows:

Step E.1: Estimation of the signal ŝ(t) on the basis of the incidences{{circumflex over (θ)}₁ . . . {circumflex over (θ)}_(M+P)} and of thesignal x(t) of equation (1) according to the method of steps C.1 and C.2described previously.

Step E.2: i=1 Ψ_(d)= and Ψ_(r)=

Step E.3: On the basis of the i-th component ŝ_(i)(t) of ŝ(t)construction of the vectors x(t)=[ŝ_(i)(t+LT_(e)) . . .ŝ_(i)(t−LT_(e))]^(T) and y(t)=x_(B)(t) and then construction of thecriterion ĉ_(xy)(τ) of equation (35).

Step E.4: Search for the P′ maxima Δτ_(k) of the criterion ĉ_(xy)(τ)such that ĉ_(xy)(Δτ_(k))>η.

Step E.5: If {circumflex over (θ)}_(i)∈{θ_(1d) . . . θ_(Md)} thiscorresponds to the presence of a direct path and Δτ_(md)=min{Δτ_(k) pour1≦k≦P′}: Ψ_(d)=Ψ_(d)∪{({circumflex over (θ)}_(i),Δτ_(md))}.

Step E.6: If {circumflex over (θ)}_(i)∈{θ_(1r) . . . θ_(Pr)} correspondsto the presence of a multi-path and Δτ_(mr)=min{Δτ_(k) for 1≦k≦P′}:Ψ_(r)=Ψ_(r)∪{({circumflex over (θ)}_(i),Δτ_(md))}.

Step E.7: i=i+1 and if then return to step E.3.

Location Module

Location of a Source (Transmitter-Obstacle) on the Basis of a Pair ofAOA-TDOA Parameters

The AOA-TDOA parameter pairs ({circumflex over (θ)}_(md),Δτ_(md)) and({circumflex over (θ)}_(pr),Δτ_(pr)) make it possible to locaterespectively the transmitters E_(m) and the reflectors (or obstacles) atR_(p). According to FIG. 11, the method must determine the position ofthe transmitter knowing that its direction of arrival is θ and that theTDOA between the two asynchronous stations A and B is Δτ. It istherefore necessary to solve the following equation system

$\begin{matrix}{{{\Delta\tau} = \frac{{{BM}} - {{AM}}}{c}}{and}{\theta = {{angle}\mspace{14mu} {\left( {{AM},{AB}} \right).}}}} & (38)\end{matrix}$

which has solution M=E_(m) according to FIG. 4. The coordinates(x_(m),y_(m)) of E_(m) then satisfy

$\begin{matrix}{{x_{m} = {x_{A} + {\frac{\left( {\Delta \; \tau \; c} \right)^{2} - {{AB}}^{2}}{2\left( {\left( {\Delta \; \tau \; c} \right) - {{{AB}}{\cos (\theta)}}} \right)}\cos \; (\theta)}}}{y_{m} = {y_{A} + {\frac{\left( {\Delta \; \tau \; c} \right)^{2} - {{AB}}^{2}}{2\left( {\left( {\Delta \; \tau \; c} \right) - {{{AB}}{\cos (\theta)}}} \right)}\sin \; {(\theta).}}}}} & (39)\end{matrix}$

where c is the speed of light, (x_(A),y_(A)) the coordinates of A and∥AB∥ the distance between A and B.

The uncertainty ellipse for the location of the transmitter at E_(m) isconstructed on the basis of a knowledge of the standard deviation σ_(Δτ)and of the mean Δ τ of the TDOA Δτ as well as of the standard deviationand of the mean θ of the estimation of the angle of incidence “AOA” θ.The parameters of this ellipse are illustrated in FIG. 12.

The equation of the uncertainty ellipse is then

x(t)=x _(m) +δD _(m) ^(max) cos(φ_(m))cos(t)−δD _(m) ^(min)sin(φ_(m))sin(t)

y(t)=y _(m) +δD _(m) ^(max) sin(φ_(m))cos(t)+δD _(m) ^(min)cos(φ_(m))sin(t).   (40)

for 0≦t≦360°. The parameters of the ellipse (δD_(m) ^(min),δD_(m)^(max),φ_(m)) are estimated on the basis of K points M_(k)(x_(k),y_(k))with coordinates (x_(k),y_(k))

$\begin{matrix}{{x_{k} = {x_{A} + {\frac{\left( {\tau_{k}c} \right)^{2} - {{AB}}^{2}}{2\left( {\left( {\tau_{k}c} \right) - {{{AB}}{\cos \left( \theta_{k} \right)}}} \right)}{\cos \left( \theta_{k} \right)}}}}{y_{k} = {y_{A} + {\frac{\left( {\tau_{k}c} \right)^{2} - {{AB}}^{2}}{2\left( {\left( {\tau_{k}c} \right) - {{{AB}}{\cos \left( \theta_{k} \right)}}} \right)}{\sin \left( \theta_{k} \right)}}}}{with}{\theta_{k} = {{\overset{\_}{\theta} + {{\cos \left( {2\pi \frac{k}{K}} \right)}\sigma_{\theta}\mspace{14mu} {and}\mspace{14mu} \tau_{k}}} = {{\Delta \overset{\_}{\tau}} + {{\sin \left( {2\pi \frac{k}{K}} \right)}{\sigma_{\tau_{m}}.}}}}}} & (41)\end{matrix}$

And finally,

$\begin{matrix}{{{\delta \; D_{m}^{\max}} = {{\max\limits_{k}\left\{ \sqrt{\left( {x_{k} - x_{m}} \right)^{2} + \left( {y_{k} - y_{m}} \right)^{2}} \right\}} = \sqrt{\left( {x_{k_{\max}} - x_{m}} \right)^{2} + \left( {y_{k_{\max}} - y_{m}} \right)^{2}}}}{{\delta \; D_{m}^{\min}} = {\min\limits_{k}\left\{ \sqrt{\left( {x_{k} - x_{m}} \right)^{2} + \left( {y_{k} - y_{m}} \right)^{2}} \right\}}}{\phi_{m} = {{angle}\mspace{14mu} \left( {\left( {x_{k_{\max}} - x_{m}} \right) + {j\left( {y_{k_{\max}} - y_{m}} \right)}} \right)}}} & (42)\end{matrix}$

To summarize, the steps of the method for locating a transmitter and/oran obstacle according to the invention are as follows:

Step No. 1: On the basis of the knowledge of the position E0 of thereference transmitter and of those of the stations at A and B, calculatethe AOA-TDOA pair (θ_(ref),Δσ_(ref)), for the transmitter E0.

Step No. 2: Initialization of the steps: k=1, Ω_(d)= and Ω_(r)=.

Step No. 3: On the basis of the sensor signals x(t) such that(k−1)T≦t<kT apply a coherent multi-path elementary goniometry accordingto the steps of the sub-method of steps D, for example, giving a set ofincidences Θ_(d)={θ_(1d) . . . θ_(Md)} associated with the direct pathsand an incidence set Θ_(r)={θ_(1r) . . . θ_(Pr)} associated with thereflected paths.

Step No. 4: On the basis of the sets Θ_(d) and Θ_(r) as well as sensorsignals x(t) such that (k−1)T≦t<kT, application of the steps of thesub-method of steps E described previously, giving a set of AOA-TDOApairs

$\Psi_{d} = {\bigcup\limits_{i}\left\{ \left( {{\hat{\theta}}_{id},{\Delta \; \tau_{id}}} \right) \right\}}$

associated with the direct paths and a set of AOA-TDOA pairs

$\Psi_{r} = {\bigcup\limits_{i}\left\{ \left( {{\hat{\theta}}_{ir},{\Delta \; \tau_{ir}}} \right) \right\}}$

associated with the reflected paths.

Step No. 5: Ω_(d)=Ω_(d)∪Ψ_(d) and Ω_(r)=Ω_(r)∪Ψ_(r)

Step No. 6: k=k+1, if k<K then return to step No. 3.

Step No. 7: On the basis of the set of data Ω_(d), extract the totalnumber M of transmitters as well as the mean and standard deviationvalues of the AOA-TDOA parameters of each of the direct paths so as toobtain

${\overset{\_}{\Omega}}_{d} = {\bigcup\limits_{i}\left\{ \left( {{{\overset{\_}{\theta}}_{md}\mspace{14mu} {and}{\,\mspace{14mu} \,_{\theta_{md}}}},{{\Delta {\overset{\_}{\tau}}_{md}\mspace{14mu} {and}\mspace{20mu} \sigma_{\Delta \; \tau_{md}}\mspace{14mu} {for}\mspace{14mu} 1} \leq m \leq M}}\mspace{11mu} \right\} \right.}$

where θ _(md) and σ_(θ) _(md) are the mean and standard deviation valuesof the incidence of the m^(th) transmitter and Δ τ _(md) and σ_(Δτ)_(md) are the mean and standard deviation values of the TDOA of thissame transmitter according to a technique known to the person skilled inthe art.

Step No. 8: On the basis of the set of data Ω_(r) extract the totalnumber P of reflectors as well as mean and standard deviation values ofthe AOA-TDOA parameters of each of the direct paths so as to obtain

${\overset{\_}{\Omega}}_{r} = {\bigcup\limits_{i}\left\{ {{\left( {{{\overset{\_}{\theta}}_{pr}\mspace{14mu} {and}\mspace{14mu} \sigma_{\theta_{pr}}},{\Delta \; {\overset{\_}{\tau}}_{pr}\mspace{14mu} {and}\mspace{14mu} \sigma_{\Delta \; \tau_{pr}}}} \right)\mspace{14mu} {for}\mspace{14mu} 1} \leq p \leq P} \right\}}$

where θ _(pr) and σ_(θ) _(pr) are the mean and standard deviation valuesof the incidence of the p^(th) reflector and

$\Delta \; {\overset{\_}{\tau}}_{r\; d}\mspace{14mu} {and}\mspace{14mu} \sigma_{\Delta \; \tau_{r\; d}}$

are the mean and standard deviation values of the TDOA of this samereflector, the extraction can be done by one of the “clustering”techniques known to the person skilled in the art. For example, it canbe carried out by applying the association method disclosed in patent FR04 11448.

Step No. 9: On the basis of the knowledge of the incidence θ_(ref) ofthe reference transmitter, search the set Ω _(d) for the incidence θ_(m) _(ref) _(d) which is closest to θ_(ref).

Step No. 10: Correct the orientation error of the antenna (sensornetworks) by performing in the sets Ω _(d) and Ω _(r): θ _(md)= θ_(md)+(θ_(ref)− θ _(m) _(ref) _(d)) and θ _(pr)= θ _(pr)+(θ_(ref)− θ_(m) _(ref) _(d)).

Step No. 11: Correct the TDOA due to the asynchronism of the receiversat A and B by performing in the sets Ω _(d) and Ω _(r): Δ τ _(md)=Δ τ_(md)+(Δτ_(ref)−Δ τ _(m) _(ref) _(d)) and Δ τ _(pr)=Δ τ_(pr)+(Δτ_(ref)−Δ τ _(m) _(ref) _(d)), since the index m_(ref)dassociated with the reference transmitter was identified in step No. 9and as the exact TDOA Δτ_(ref) of the reference transmitter wascalculated in step No. 1

Step No. 12: On the basis of the knowledge of the mean incidences θ_(mr) and θ _(pr) and then of the level of the calibration errors,calculate for each of the sources according to [5] for example standarddeviations σ₀ _(md) ^(cal) and σ₀ _(pr) ^(cal) related to thecalibration errors.

Step No. 13: Correct the standard deviations of the TDOAs by takingaccount of the calibration errors in the sets Ω _(d) and Ω _(r): σ_(θ)_(md) =σ_(θ) _(md) +σ_(θ) _(md) ^(cal) and σ_(θ) _(pr) =σ_(θ) _(pr)+σ_(θ) _(pr) ^(cal).

Step No. 14: On the basis of the pairs ( θ _(md),Δ τ _(md)) and ( θ_(pr),Δ τ _(pr)), determine the positions of the transmitters E_(m)( x_(md), y _(md)) and of the reflectors R_(p)( x _(pr), y _(pr)) byapplying the calculation of equation (39).

Step No. 15: On the basis of the pairs ( θ _(md) and σ_(θ) _(md) ,Δ τ_(md) and σ_(Δτ) _(md) ) and ( θ _(pr) and σ_(θ) _(pr) ,Δ τ _(pr) andσ_(Δτ) _(pr) ), determine the positions of the uncertainty ellipses forthe transmitters with position E_(m) and for the reflectors withposition R_(p) by applying, for example, the calculation of equations(41)(42).

The invention makes it possible to locate several transmitters. It takesinto account the presence of multi-paths by giving the position of thereflectors. It does not make any assumption about the transmittedsignals: they can be of different bands, with or without pilot signals.The signals can equally well be radio-communication signals or RADARsignals.

1. A method for locating one or more transmitters Ei in the potentialpresence of obstacles Rp in a network comprising at least one firstreceiving station A and one second receiving station B asynchronous withA comprising the following steps: the identification of a referencetransmitter of known position E₀ by a calculation of the AOA-TDOA pair(θ_(ref),Δτ_(ref)) on the basis of the knowledge of the position E0 ofthe reference transmitter and of those of the stations at A and B; anestimation of the direction of arrival of the transmitter ortransmitters and of the reflectors (or estimation of the AOA) on thefirst station A; the separation of the signals received on the firststation A by spatial filtering in the direction of the source(transmitters and/obstacles); the separation of the incidencesoriginating from the transmitters from those originating from theobstacles by inter-correlating the signals arising from the spatialfiltering at A; the estimation of the time difference of arrival or TDOAof a source (transmitters and/obstacles) by inter-correlating the signalof the source (transmitters and/obstacles) received at A with thesignals received on the second receiving station B: for each transmittersource Ei (or obstacles Rj) a pair (AOA, TDOA) is then obtained; asynthesis of the measurements of the pairs (AOAi, TDOAi) of each source(Ei, Rp) so as to enumerate the sources and to determine the means andstandard deviation of their AOA and TDOA parameters; the determinationof the error of synchronism between the receiving stations A and B byusing the reference transmitter E₀, and then the correction of thiserror on all the TDOAi of the pairs (AOAi, TDOAi) arising from thesynthesis; the determination of the orientation error of the receivingstation A by using the reference transmitter E₀, and then the correctionof this error on all the AOAi of the pairs (AOAi, TDOAi) arising fromthe synthesis; and the location of the various transmitters on the basisof each pair (AOAi, TDOAi).
 2. The method as claimed in claim 1,comprising at least one step in which an uncertainty ellipse for themeasurements of standard deviation of the parameters (AOA, TDOA) isestablished.
 3. The method as claimed in claim 1, comprising thefollowing steps: Step No. 2: Initialization of the steps: k=1, Ω_(d)=and Ω_(r)=. Step No. 3: On the basis of the sensor signals x(t) suchthat (k−1)T≦t<kT application of an elementary goniometry, giving a setof incidences Θ_(d)={θ_(1d) . . . θ_(Md)} associated with the directpaths and an incidence set Θ_(r)={θ_(1r) . . . θ_(Pr)} associated withthe reflected paths. Step No. 4: On the basis of the sets Θ_(d) andΘ_(r) as well as sensor signals x(t) such that (k−1)T≦t<kT, apply ascheme for associating the angles of arrival and the TDOAs so as toobtain a set of AOA-TDOA pairs$\Psi_{d} = {\bigcup\limits_{i}\left\{ \left( {{\hat{\theta}}_{id},{\Delta \; \tau_{id}}} \right) \right\}}$associated with the direct paths and a set of AOA-TDOA pairs$\Psi_{r} = {\bigcup\limits_{i}\left\{ \left( {{\hat{\theta}}_{ir},{\Delta \; \tau_{ir}}} \right) \right\}}$associated with the reflected paths. Step No. 5: Ω_(d)=Ω_(d)∪Ψ_(d) andΩ_(r)=Ω_(r)∪Ψ_(r) Step No. 6: k=k+1, if k<K then return to step No. 3.Step No. 7: On the basis of the set of data Ω_(d), extract the totalnumber M of transmitters as well as the mean and standard deviationvalues of the AOA-TDOA parameters of each of the direct paths so as toobtain${\overset{\_}{\Omega}}_{d} = {\bigcup\limits_{i}\left\{ {{\left( {{{\overset{\_}{\theta}}_{md}\mspace{14mu} {and}\mspace{14mu} \sigma_{\theta_{md}}},{\Delta {\overset{\_}{\tau}}_{md}\mspace{14mu} {and}\mspace{14mu} \sigma_{\Delta \; \tau_{md}}}} \right)\mspace{14mu} {for}\mspace{14mu} 1} \leq m \leq M} \right\}}$where θ _(md) and σ₇₄ _(md) are the mean and standard deviation valuesof the incidence of the m^(th) transmitter and Δ τ _(md) et σ_(Δr) _(md)are the mean and standard deviation values of the TDOA of this sametransmitter according to a technique known to the person skilled in theart. Step No. 8: On the basis of the set of data Ω_(r) extract the totalnumber P of reflectors as well as mean and standard deviation values ofthe AOA-TDOA parameters of each of the direct paths to obtain${\overset{\_}{\Omega}}_{r} = {\bigcup\limits_{i}\left\{ {{\left( {{{\overset{\_}{\theta}}_{pr}\mspace{14mu} {and}\mspace{14mu} \sigma_{\theta_{pr}}},{\Delta {\overset{\_}{\tau}}_{pr}\mspace{14mu} {and}\mspace{14mu} \sigma_{\Delta \; \tau_{pr}}}} \right)\mspace{14mu} {for}\mspace{14mu} 1} \leq p \leq P} \right\}}$where θ _(pr) and σ_(θ) _(pr) are the mean and standard deviation valuesof the incidence of the p^(th) reflector and Δ τ _(rd) and σ_(Δτ) _(ref)are the mean and standard deviation values of the TDOA of this samereflector Step No. 9: On the basis of the knowledge of the incidenceθ_(ref) of the reference transmitter, search the set Ω _(d) for theincidence θ _(m) _(ref) _(d) which is the closest to θ_(ref). Step No.10: Correct the orientation error of the antenna by performing in thesets Ω _(d) and Ω _(r): θ _(md)= θ _(md)+(θ_(ref)− θ _(m) _(ref) _(d))and θ _(pr)= θ _(pr)+(θ_(ref)− θ _(m) _(ref) _(d)). Step No. 11: Correctthe TDOA error due to the asynchronism of the receivers at A and B byperforming in the sets Ω _(d) and Ω _(r): Δ τ _(md)=Δ τ_(md)+(Δτ_(ref)−Δ τ _(m) _(ref) _(d)) and Δ τ _(pr)=Δ τ_(pr)+(Δτ_(ref)−Δ τ _(m) _(ref) _(d)). Step No. 12: On the basis of theknowledge of the mean incidences θ _(mr) and θ _(pr) and then of thelevel of the calibration errors, calculate, for each of the sources,standard deviations σ_(θ) _(md) ^(cal) et σ_(θ) _(pr) ^(cal) related tothe calibration errors, Step No. 13: Correct the values of standarddeviations of the TDOAs by performing in the sets Ω _(d) and Ω _(r):σ_(θ) _(md) =σ_(θ) _(md) +σ_(θ) _(md) ^(cal) and σ_(θ) _(pr) =σ_(θ)_(pr) +σ_(θ) _(pr) ^(cal). Step No. 14: On the basis of the pairs ( θ_(md),Δ τ _(md)) and ( θ _(pr),Δ τ _(pr)), determine the positions ofthe transmitters E_(m)( x _(md), y _(md)) and of the reflectors R_(p)( x_(pr), y _(pr)) Step No. 15: On the basis of the pairs ( θ _(md) andσ_(θ) _(md) ,Δ θ _(md) and σ_(Δτ) _(md) ) and ( θ _(pr) and σ_(θ) _(pr),Δ τ _(pr) and σ_(Δτ) _(pr) ), determine the positions of theuncertainty ellipses for the transmitters with position E_(m) and forthe reflectors with position R_(p).
 4. A system for locating one or moretransmitters Ei in the potential presence of obstacles Rp in a networkcomprising at least one first receiving station A and one secondreceiving station B asynchronous with A wherein the system comprises atleast one reference transmitter E₀ whose position is known and aprogrammed processor and memory containing instructions for implementingthe following steps: the identification of a reference transmitter ofknown position E₀ by a calculation of the AOA-TDOA pair(θ_(ref),Δτ_(ref)) on the basis of the knowledge of the position E0 ofthe reference transmitter and of those of the stations at A and B: anestimation of the direction of arrival of the transmitter ortransmitters and of the reflectors (or estimation of the AOA) on thefirst station A: the separation of the signals received on the firststation A by spatial filtering in the direction of the source(transmitters and/obstacles): the separation of the incidencesoriginating from the transmitters from those originating from theobstacles by inter-correlating the signals arising from the spatialfiltering at A: the estimation of the time difference of arrival or TDOAof a source (transmitters and/obstacles) by inter-correlating the signalof the source (transmitters and/obstacles) received at A with thesignals received on the second receiving station B: for each transmittersource Ei (or obstacles Rj) a pair (AOA, TDOA) is then obtained: asynthesis of the measurements of the pairs (AOAi, TDOAi) of each source(Ei, Rp) so as to enumerate the sources and to determine the means andstandard deviation of their AOA and TDOA parameters: the determinationof the error of synchronism between the receiving stations A and B byusing the reference transmitter E₀, and then the correction of thiserror on all the TDOAi of the pairs (AOAi, TDOAi) arising from thesynthesis: the determination of the orientation error of the receivingstation A by using the reference transmitter E₀, and then the correctionof this error on all the AOAi of the pairs (AOAi, TDOAi) arising fromthe synthesis; and the location of the various transmitters on the basisof each pair (AOAi, TDOAi).